Legendre transform: Difference between revisions
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The '''Legendre transform''' (Adrien-Marie Legendre) | |||
is used to perform a change ''change of variables'' | |||
(see, for example, Ref. 1, Chapter 4 section 11 Eq. 11.20 - 11.25): | |||
If one has the function <math>f(x,y);</math> one can write | |||
:<math>df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy</math> | |||
Let <math>p= \partial f/ \partial x</math>, and <math>q= \partial f/ \partial y</math>, thus | |||
:<math>df = p~dx + q~dy</math> | |||
If one subtracts <math>d(qy)</math> from <math>df</math>, one has | |||
:<math>df- d(qy) = p~dx + q~dy -q~dy - y~dq</math> | |||
or | |||
:<math>d(f-qy)=p~dx - y~dq </math> | |||
Defining the function <math>g=f-qy</math> | |||
then | |||
:<math>dg = p~dx + q~dy</math> | |||
The partial derivatives of <math>g</math> are | |||
:<math>\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y</math>. | |||
==Example== | |||
==References== | |||
#Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition. | |||
Revision as of 11:07, 28 May 2007
The Legendre transform (Adrien-Marie Legendre) is used to perform a change change of variables (see, for example, Ref. 1, Chapter 4 section 11 Eq. 11.20 - 11.25):
If one has the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y);} one can write
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy}
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p= \partial f/ \partial x} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q= \partial f/ \partial y} , thus
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df = p~dx + q~dy}
If one subtracts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(qy)} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df} , one has
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df- d(qy) = p~dx + q~dy -q~dy - y~dq}
or
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f-qy)=p~dx - y~dq }
Defining the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=f-qy} then
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dg = p~dx + q~dy}
The partial derivatives of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y} .
Example
References
- Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition.